Finger Competition Research Articles

Geometric approach to viscous fingering on a cone

We study fluid flow and the formation of viscous fingering patterns on a two-dimensional conical background space, defined as the conical Hele–Shaw cell. We approach the problem geometrically and study how the nontrivial topological structure of the conical cell affects the evolution of the interface separating two viscous fluids. We perform a perturbative weakly nonlinear analysis of the problem and derive a mode-coupling differential equation which describes fluid–fluid interface behaviour. Our nonlinear study predicts the formation of fingering structures in which fingers of different lengths compete and split at their tips. The shape of the emerging patterns show a significant sensitivity to variations in the cell's topological features, which can be monitored by changing the cone opening angle. We find that for increasingly larger values of the opening angle, finger competition is inhibited while finger tip-splitting is enhanced.

Effects of small surface tension in Hele-Shaw multifinger dynamics: an analytical and numerical study.

We study the singular effects of vanishingly small surface tension on the dynamics of finger competition in the Saffman-Taylor problem, using the asymptotic techniques described by Tanveer [Philos. Trans. R. Soc. London, Ser. A 343, 155 (1993)] and Siegel and Tanveer [Phys. Rev. Lett. 76, 419 (1996)], as well as direct numerical computation, following the numerical scheme of Hou, Lowengrub, and Shelley [J. Comput. Phys. 114, 312 (1994)]. We demonstrate the dramatic effects of small surface tension on the late time evolution of two-finger configurations with respect to exact (nonsingular) zero-surface-tension solutions. The effect is present even when the relevant zero-surface-tension solution has asymptotic behavior consistent with selection theory. Such singular effects, therefore, cannot be traced back to steady state selection theory, and imply a drastic global change in the structure of phase-space flow. They can be interpreted in the framework of a recently introduced dynamical solvability scenario according to which surface tension unfolds the structurally unstable flow, restoring the hyperbolicity of multifinger fixed points.

Dynamical systems approach to Saffman-Taylor fingering: dynamical solvability scenario.

A dynamical systems approach to competition of Saffman-Taylor fingers in a Hele-Shaw channel is developed. This is based on global analysis of the phase space flow of the low-dimensional ordinary-differential-equation sets associated with the classes of exact solutions of the problem without surface tension. Some simple examples are studied in detail. A general proof of the existence of finite-time singularities for broad classes of solutions is given. Solutions leading to finite-time interface pinchoff are also identified. The existence of a continuum of multifinger fixed points and its dynamical implications are discussed. We conclude that exact zero-surface tension solutions taken in a global sense as families of trajectories in phase space are unphysical because the multifinger fixed points are nonhyperbolic, and an unfolding does not exist within the same class of solutions. Hyperbolicity (saddle-point structure) of the multifinger fixed points is argued to be essential to the physically correct qualitative description of finger competition. The restoring of hyperbolicity by surface tension is proposed as the key point to formulate a generic dynamical solvability scenario for interfacial pattern selection.

Dynamics of finger arrays in a diffusion-limited growth model with a drift

We propose a phenomenological model of fingering dynamics in the presence of an external drift, motivated by recent experiments on quasi-two-dimensional electrodeposition with hydrodynamic convection. We study the dynamical transition between a Laplacian growth regime and a weak finger competition regime. The model proposed defines a wavelength-selection problem of the finger array which is coupled in a nontrivial way to the single-finger selection problem in a channel. The wavelength-selection mechanism proposed is associated to the crossover from a diffusive to a ballistic behavior of the aggregating particles, as the coarsening spatial scale of the pattern matches the appropriate diffusion length. This yields specific predictions for the selected wavelength. The dynamics of the model are studied numerically with boundary-integral methods in the quasistatic approximation and, complementarily, using a biased random walk Monte Carlo scheme on a lattice. Results are in qualitative agreement with the theoretical predictions.

Nonlinear effects due to gravity in a conical Hele-Shaw cell.

In this work we study the viscous fingering instability in a conical Hele-Shaw cell under the presence of gravity. We focus on understanding how the dynamical evolution of the fingering patterns is affected by the combined action of gravity and cell topology. Gravity-induced nonlinear effects are studied by a mode-coupling approach. Our results show that the interplay between gravity and cell topology leads to important effects, and profoundly modifies pattern evolution. We have found that the most dramatic consequences refer to finger tip behavior. Depending on the relative values of fluids' densities and viscosities, finger tip splitting reaches maximum intensity at well defined, preferred values of the cell opening angle. In fact, finger tip splitting can be completely replaced by finger tip sharpening as the cell angle is varied. Finger competition dynamics is also significantly changed: it is considerably enhanced (restrained) if the displaced fluid is more (less) dense.

Analytical approach to viscous fingering in a cylindrical Hele-Shaw cell.

We report analytical results for the development of the viscous fingering instability in a cylindrical Hele-Shaw cell of radius a and thickness b. We derive a generalized version of Darcy's law in such cylindrical background, and find it recovers the usual Darcy's law for flow in flat, rectangular cells, with corrections of higher order in b/a. We focus our interest on the influence of the cell's radius of curvature on the instability characteristics. Linear and slightly nonlinear flow regimes are studied through a mode-coupling analysis. Our analytical results reveal that linear growth rates and finger competition are inhibited for an increasingly larger radius of curvature. The absence of tip-splitting events in cylindrical cells is also discussed.

Dynamics and selection of fingering patterns. Recent developments in the Saffman–Taylor problem

We study singular effects of surface tension in the dynamics of the finger competition in the Saffman–Taylor problem with channel geometry. First, we study in detail some relevant classes of exact solutions in the absence of surface tension and compare them to finite surface tension. We conclude that (nonsingular) zero-surface tension solutions are generically unphysical. We show that the elementary two-finger competition process in the absence of surface tension is structurally unstable and this fact is ultimately responsible for the lack of genuine finger competition. Second, we generalize solvability theory to study selection of multifinger configurations with finite surface tension. We find that a discrete set of nontrivial multifinger solutions, with stationary coexistence of unequal fingers, are selected by surface tension out of a continuum of solutions. We discuss the implications of these results on a possible dynamic solvability scenario of selection.

Laplacian Growth I: Finger Competition and Formation of a Single Saffman-Taylor Finger without Surface Tension: An Exact Result

We study the exact non-singular zero-surface tension solutions of the Saffman-Taylor problem for all times. We show that all moving logarithmic singularities a_k(t) in the complex plane \omega = e^{i\phi}, where \phi is the stream function, are repelled from the origin, attracted to the unit circle and eventually coalesce. This pole evolution describes essentially all the dynamical features of viscous fingering in the Hele-Shaw cell observed by Saffman and Taylor [Proc. R. Soc. A 245, 312 (1958)], namely tip-splitting, multi-finger competition, inverse cascade, and subsequent formation of a single Saffman-Taylor finger.

Radial fingering in a Hele-Shaw cell: a weakly nonlinear analysis

The Saffman-Taylor viscous fingering instability occurs when a less viscous fluid displaces a more viscous one between narrowly spaced parallel plates in a Hele-Shaw cell. Experiments in radial flow geometry form fan-like patterns, in which fingers of different lengths compete, spread and split. Our weakly nonlinear analysis of the instability predicts these phenomena, which are beyond the scope of linear stability theory. Finger competition arises through enhanced growth of sub-harmonic perturbations, while spreading and splitting occur through the growth of harmonic modes. Nonlinear mode-coupling enhances the growth of these specific perturbations with appropriate relative phases, as we demonstrate through a symmetry analysis of the mode coupling equations. We contrast mode coupling in radial flow with rectangular flow geometry.

Surface tension and dynamics of fingering patterns

We study the minimal class of exact solutions of the Saffman-Taylor problem with zero surface tension, which contains the physical fixed points of the regularized (non-zero surface tension) problem. New fixed points are found and the basin of attraction of the Saffman-Taylor finger is determined within that class. Specific features of the physics of finger competition are identified and quantitatively defined, which are absent in the zero surface tension case. This has dramatic consequences for the long-time asymptotics, revealing a fundamental role of surface tension in the dynamics of the problem. A multifinger extension of microscopic solvability theory is proposed to elucidate the interplay between finger widths, screening and surface tension.

Finger competition and viscosity contrast in viscous fingering. A topological approach

Abstract Finger competition in rectangular Hele-Shaw cells is discussed with particular focus on topological features of the velocity field. A reduced description of the interface dynamics is provided in terms of the motion of topological defects which obey simple conservation rules. Based on the existence of a topological invariant, a time-dependent quantity S(t) , directly related to the number of topological defects, is proposed as a global characterization of the finger competition. This quantity is not simply related to the interface shape but may be used as a measure of the ‘complexity’ of evolving patterns and to elucidate the conditions for eventual approach to a single finger stationary solution. Some representative numerical simulations for two extreme limits of the viscosity contrast parameter are presented and analyzed in terms of the dynamics of topological defects. Two specific sequences or histories for the creation/annihilation of defects are then identified, corresponding to radically different behavior of competing fingers. With the help of a local vorticity analysis, we conclude that, in the limit of low viscosity contrast, the Saffman-Taylor finger has a drastically reduced but non-vanishing basin of attraction.

Finger competition and viscosity contrast in viscous fingering. A topological approach

Finger competition in rectangular Hele-Shaw cells is discussed with particular focus on topological features of the velocity field. A reduced description of the interface dynamics is provided in terms of the motion of topological defects which obey simple conservation rules. Based on the existence of a topological invariant, a time-dependent quantity S(t), directly related to the number of topological defects, is proposed as a global characterization of the finger competition. This quantity is not simply related to the interface shape but may be used as a measure of the ‘complexity’ of evolving patterns and to elucidate the conditions for eventual approach to a single finger stationary solution. Some representative numerical simulations for two extreme limits of the viscosity contrast parameter are presented and analyzed in terms of the dynamics of topological defects. Two specific sequences or histories for the creation/annihilation of defects are then identified, corresponding to radically different behavior of competing fingers. With the help of a local vorticity analysis, we conclude that, in the limit of low viscosity contrast, the Saffman-Taylor finger has a drastically reduced but non-vanishing basin of attraction.

Removing the stiffness from interfacial flows with surface tension

A new formulation and new methods are presented for computing the motion of fluid interfaces with surface tension in two-dimensional, irrotational, and incompressible fluids. Through the Laplace-Young condition at the interface, surface tension introduces high-order terms, both nonlinear and nonlocal, into the dynamics. This leads to severe stability constraints for explicit time integration methods and makes the application of implicit methods difficult. This new formulation has all the nice properties for time integration methods that are associated with having a linear, constant coefficient, highest order term. That is, using this formulation, we give implicit time integration methods that have no high order time step stability constraint associated with surface tension and are explicit in Fourier space. The approach is based on a boundary integral formulation and applies more generally, even to problems beyond the fluid mechanical context. Here they are applied to computing with high resolution the motion of interfaces in Hele-Shaw flows and the motion of free surfaces in inviscid flows governed by the Euler equations. One Hele-Shaw computation shows the behavior of an expanding gas bubble over long-time as the interface undergoes successive tip-splittings and finger competition. A second computation shows the formation of a very ramified interface through the interaction of surface tension with an unstable density stratification. In Euler flows, the computation of a vortex sheet shows its roll-up through the Kelvin-Helmholtz instability. This motion culminates in the late time self-intersection of the interface, creating trapped bubbles of fluid. This is, we believe, a type of singularity formation previously unobserved for such flows in 2D. Finally, computations of falling plumes in an unstably stratified Boussinesq fluid show a very similar behavior.

INTERFACE EQUATION AND VISCOSITY CONTRAST IN HELE-SHAW FLOW

We derive an integro-differential equation for the evolution of the interface separating two immiscible viscous fluids in a Hele-Shaw cell with a channel geometry, for arbitrary viscosity contrast. Our equation differs from a previous one obtained by a vortex-sheet formulation of the problem, in that the normal component of the interface velocity is formally decoupled from the gauge-dependent tangential part. The result is thus a closed integral equation for the normal velocity. We briefly comment on the advantages of such a formulation and implement an alternative computational algorithm based on it. Preliminary numerical results confirm a highly inefficient finger competition in the zero viscosity contrast limit.